The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.

In this thesis, high order accurate discretization schemes for partial differential equations are investigated. In the first paper, the linearized two-dimensional Navier-Stokes equations are considered. A special formulation of the boundary conditions is used and estimates for the solution to the continuous problem in terms of the boundary conditions are derived using a normal mode analysis. Similar estimates are achieved for the discretized equations. For the discretization, a second order finite difference scheme on a staggered mesh is used. In Paper II, the analysis for the second order scheme is used to develop a fourth order scheme for the fully nonlinear Navier-Stokes equations. The fully nonlinear incompressible Navier-Stokes equations in two space dimensions are considered on an orthogonal curvilinear grid. Numerical tests are performed with a fourth order accurate Padé type spatial finite difference scheme and a semi-implicit BDF2 scheme in time. In Papers III-V, a class of high order accurate time-discretization schemes based on the deferred correction principle is investigated. The deferred correction principle is based on iteratively eliminating lower order terms in the local truncation error, using previously calculated solutions, in each iteration obtaining more accurate solutions. It is proven that the schemes are unconditionally stable and stability estimates are given using the energy method. Error estimates and smoothness requirements are derived. Special attention is given to the implementation of the boundary conditions for PDE. The scheme is applied to a series of numerical problems, confirming the theoretical results. In the sixth paper, a time-compact fourth order accurate time discretization for the one- and two-dimensional wave equation is considered. Unconditional stability is established and fourth order accuracy is numerically verified. The scheme is applied to a two-dimensional wave propagation problem with discontinuous coefficients.